A Hard Day's Night by numbers: The Beatles decoded

5 Jul 2012, 2:00 pm   -   Source: The Conversation
Share This
0 
+ Comment
10

The Conversation

(AAP)

(AAP)

Jason Brown from Dalhousie University believes he has found the maths behind some of The Beatles' songs.

By Jason Brown, Dalhousie University

“TWANG! It’s been a …”

There is perhaps no song as quintessentially Beatle-ish as A Hard Day’s Night – it just bubbles with unbridled enthusiasm and joy. And in my mind, there’s no other opening chord of a rock song that is as instantly recognisable as that one.

I grew up grudgingly playing the piano, practising only the half-hour before my lesson each week. But as soon as heard my first Beatles’ record, I dropped the piano to teach myself guitar eight hours a day during my high school summers.

Something about the early Beatles' music struck a chord, so to speak, deep down inside of me, and it hasn’t left.

At about the same time, my love for mathematics blossomed, and I played in a band while attending my undergraduate studies. It was a tough choice, but I gave up music for the safer gig, as a mathematician. But unbeknown to me, the music that lay dormant inside me would serendipitously mix with the math inside me.

In 2004 I heard it was the 40th anniversary of the Beatles’ first movie – A Hard Day’s Night – and the soundtrack of the same name. All of the media attention brought to mind that famous opening chord that opened the movie and title song.

While teaching myself guitar years earlier, I had invested in a lot of Beatles songbooks, only to find that every book had a different transcription for how the author thought George Harrison had coaxed that initial sound out of his brand new twelve-string Rickenbacker guitar. All were derived by some combination of listening and music theory, but to me none sounded quite right.

1999 Rickenbacker guitar with the distinctive R-tailpiece. Mr.Fingers

My mathematical outlook had me take a different approach in 2004 – was there a scientific way to decide how the chord was played? Indeed, I had read a math book for leisure (yes we mathematicians do that sort of thing!) about ten years earlier that described the mathematics of sound and music.

In particular, there was a process, called a Fourier transform, that could allow one to decompose a sound wave into its constitute pure tones (which were modelled by sine and cosine curves). I had also remembered that there were algorithms to do just that, so I embarked on some CSI-like musical forensics. I took a small part of the opening chord and ran it through a Fourier transform, and held my breath waiting for the output.

The spectrum of the 48 frequencies of largest amplitude from "The Chord". Jason Brown

It was a bit daunting – there were thousands of frequencies in the opening chord. But all was not lost, as I could tell the amplitudes of the frequencies, and the amplitude corresponds roughly with the loudness. So I began to make mathematical deductions from the data, and I quickly came upon some interesting conclusions.

First, all of the transcriptions I had seen for the guitar chord were incorrect – they had a low G note present, and the mathematics clearly indicated that the frequency simply wasn’t present.

Musicians thought they had heard the note, and as the key of the song was G, they believed it to be there all the more strongly. But it wasn’t.

Furthermore, I could see that the frequencies often were not particularly close to notes, so that it would have behoved the Beatles’ producer, George Martin, to have knocked on the studio window before the final take of the song and said: “Better tune up again, boys.” The Beatles’ guitars were gloriously slightly out-of-tune, adding to the difficulty in reproducing the chord.

The Beatles with producer George Martin in the studio at Abbey Road. Capitol Records

A much bigger problem loomed. There were three frequencies corresponding to a certain “F” note, with no corresponding note up the octave, and this meant that note couldn’t have been played on George Harrison’s twelve string, and further, there was no way for the Beatles’ guitars to cover the frequencies. The answer involved throwing out the assumption that only the Beatles played on the opening chord.

The correct chords to play discovered via the Fourier transform Jason Brown
A solution lay with insertion of a piano into the mix, as pianos have, toward the top end of the keyboard, three identically tuned strings under each note. Upon this realisation, the remainder of the chord began to unravel fairly quickly, and I could deduce what instruments (guitars, bass and piano) played what notes. A little bit of math went a long way!

The greatest difficulty I encountered after the research was finding a public forum to publish the work. It was going to appear in a peer-reviewed journal, but I thought the story was interesting enough for everyone to read. One magazine refused to read it based on the fact that the article had mathematics in it! But Guitar Player magazine loved the work, and was happy to publish the article, and the rest is, as they say, history.

Over the ensuing years, I have applied mathematics in a variety of ways to analyse pop music. In a second article in Guitar Player magazine I deduced mathematically that George Harrison must have recorded his famous, brilliant solo in A Hard Day’s Night by slowing down the tape speed in half, and recording the solo at half-speed down the octave.

Some musicians I’ve spoken with have been upset at the research, as perhaps it showed George’s technical skills were not what they should have been, but the truth I think says more – it showed George was a musician first, doing what it took to play what was in his head rather than in his fingers, and he had to have an incredible amount of confidence to choose to record a solo at half-speed, knowing that all of the world would be watching for when he played it up to speed, live (which, of course, he did!).

I’ve also written about why the music to I Want To Hold Your Hand was so imaginative and clever that it brought America to its knees, and why Paul McCartney so correctly named Little Richard’s Long Tall Sally as perhaps one of the greatest rock songs ever (and more generally, a mathematical basis for why the blues chord progression is so damn good).

Finally, in recent work with Robert Dawson of St. Marys University, we explained mathematically why George Martin’s famous edit in Strawberry Fields Forever never quite satisfied Paul (and never could).

Moreover, the research continues to open doors for me, especially as an ambassador for mathematics. I’ve written a book for the general public called Our Days Are Numbered: How Mathematics Orders Our Lives and published my first CD, Songs in the Key of Pi, of my own songs.

In fact, the Wall Street Journal came a-knocking back in 2008, and shot a video of a song I wrote in the style of the early Beatles, using mathematical principles I gleaned from their music.

And I continue to travel worldwide, giving public lectures on mathematics and music, most often with a guitar slung over my shoulder and with a rockin’ band behind me. The Beatles, it seems, gave me a great ticket to ride!

Jason Brown receives funding from the Natural Sciences and Engineering Research of Canada. He is affiliated with Dalhousie University in Halifax, Canada.

The Conversation

The Conversation , The Beatles

  • Follow us: 
  • Follow us on Facebook
  • Follow us on Twitter
  • Follow us on Youtube

Your Comments

Display: 20 | 40 | All comments per page
Previous 10 | Page 1 | Next 10

Not even close!

Sid - from NY USA, 11 months ago

Drasil's comment (first one at bottom) is spot on... Bachman's recreation of the chord (YouTube it) is DEFINITELY what is played on the record. Kind of embarrassing that Dr Brown didn't bother to "review the literature," as it were, since Bachman and Giles took it apart in 2010...

George Rocks!

Cathy - from Los Angeles, 11 months ago

All I can say is given all the technical observations it is a FANTASTIC song intro. As you can see from the 'Living in the Material World' documentary, George enjoyed experimenting with sounds and chords. He ventured into musical territories fearlessly. Thanks for the fun article!

Bad maths

Fourier - from Sydney, 11 months ago

How can you be sure that the Fourier decomposition is not picking up harmonics (higher and softer components of the individual notes which give the notes their timbre)? The Fourier transform is usually used to analyse harmonics rather than fundamental notes. Also, harmonics are not all 'in tune' with respect to the standard Western major scale. This could explain your problem with the tuning. It could also explain why you are finding notes which the instruments cannot play.

Listening is a skill not taught

Ed Sullivan - from Syracuse, NY, 11 months ago

Sorry, but yes there is a piano in that chord.

guitarist

Gringo - from Columbia South Carolina, 12 months ago

In those days, the beginnings and endings of songs were things I tended to organize," said George Martin. "We needed something striking, to be a sudden jerk into the song." At the session, Lennon played around with some fingerings for the opening chord. "It was by chance that he struck the right one," said Martin. "We knew it when we heard it." (In a February 2001 interview, Harrison said the chord is an "F with a G on top, but you'll have to ask Paul about the bass note to get the proper story." McCartney played a high D.)

its always about the first chord

Randy - from Ossining NY, 12 months ago

Has anyone noticed that the last notes of a Hard Day's night are very close to the 5 notes from Close Encounters of the third kind? just an observation

Misleading words

Me - from Not USA, 12 months ago

"Science doesn't always have the answers, it seems..." well, its purpose is to look for the answers

nice try though...

gman in NYC - from New York, 12 months ago

Bachman detailed his findings and recreated the chord exactly (as drasil said). He made an audio recording and it's available in an online search. A perfect match. I guess you didn't take into account that Harrison was playing a 12-string. The Beatles never cared much for math anyway... ;-)

Beatle's Hard Day's Night Open Chord

Dr. Terry Allen - from Spokane, WA, 12 months ago

There is no way to define the guitar position for a note without knowing how the guitar is tuned. Take pitch value of the notes and play the notes in every possible tuning and key on guitar until you find the correct positions. For 6-string chord opening Beatles' Hard Day's NIght the Turing Machine Stops: Open D Tuning (detuned 2 steps to match record) 0 7 5 4 3 5 with position vector 0 0 5 3 0 0. Open D fret string 3 fret 5 and string 4 fret 3, strings 1,2, 5, 6 open. DAGAAD = CGFGGC Wham!

good shot, mostly inaccurate

drasil - from USA, 12 months ago

Professor Brown, sadly, most of your results are inaccurate. Randy Bachman deconstructed the chord with Giles Martin at Abbey Road, straight from the isolated tracks. There is no piano in the chord. The three instruments are George's 12-string playing an oddly-voiced F with a G in the bass AND on top (which you incorrectly state does not exist); John's 6-string playing a Dsus4, and Paul's bass playing a D note. That's all there is to it. Science doesn't always have the answers, it seems...

Display: 20 | 40 | All comments per page
Previous 10 | Page 1 | Next 10

Join the Discussion

Name
City / Suburb E.g. Artarmon, Sydney
Title
Comment
You have characters remaining.
Validation
What's this?
This is a captcha-picture. It is used to prevent mass-access by robots.
All submitted comments become the property of SBS. They are moderated, so we reserve the right to edit comments and remove HTML tags. Not all submitted comments will be published. Publication does not mean we endorse the opinions expressed. Please read our terms and conditions for more information.